This isn't really an attack unique to Proof of Stake. If a node goes offline they can lose rewards or even in rare cases have their coins slashed to some extent but that isn't inherent to a Proof of Stake overall. A decent number of Proof of Stake systems instead place reward penalties on pools/nodes that go offline. The idea being that it is a penalty for not maintaining sufficient infrastructure while also not being so severe that it could be leveraged in such an attack.
Most PoS algorithms I've seen instead reserve stake slashing as a penalty for malicious behaviour. Going offline isn't by any means inherently malicious. There are however plenty of actively malicious actions that can be detected and reacted against. Often for the more severe penalties it will require some level of community involvement in the recovery stage to limit opportunities for abuse.
Additionally, it shouldn't be easy to take a block producer offline and Stake Pool(or node) Operators should be preparing for these types of attacks. I've been watching some of the work being done in the Cardano Stake Pool Operator community and the various SPO guilds have decently sophisticated architectures. "Nodes"/"Pools" are broken up into Relays, Producers, and sometimes additionally Key Generators. Key Generators produce the periodically expiring KES keys and pass them to the Producers on a schedule (to minimise potential attack surfaces). The Producers actually engage in the consensus using the keys provided by the key generators and communicate through the relays. The Relays handle the throughput and communication. This allows the producers (and by extension the key generators if used) to be largely shielded from the open net. This also allows producers and relays to have a certain amount of redundancy/failover. An architecture like that may cost more (and eat into rewards a bit more) however they are far more difficult to DDoS or compromise.
Since the barrier for the hardware is so low, a 1x2x2 or 1x2x3 (keygen x producer x relay) architecture can still be more than profitable (retaining 25% to 75% of the SPO rewards as profit). Additionally this has the advantage that various other income streams can be integrated in (state channel operation, compute nodes, storage nodes, etc) over time and the operation can be scaled up without compromising security or requiring a significant re-architecture.
Proof of Stake can be just as secure as Proof of Work but it requires that the incentives be structured properly and sufficiently hedged against potential risks.